A scientist is analyzing a function that models the temperature fluctuation during the day, given by \(f(x)= (x-4)^2 + 1\) for \(x\) in the interval [0, 8] hours. Due to a change in the experimental setup, the function undergoes a transformation: it is first reflected across the vertical line \(x=4\) and then translated 2 units to the right.
(a) Write the expression for the transformed function \(g(x)\).
(b) Determine the new coordinates of the vertex of \(f(x)\) after the transformation.
(c) Discuss how the domain of the original function is affected by these transformations.
Question
Write the expression for the transformed function \(g(x)\), given that \(f(x)= (x-4)^2 + 1\) is first reflected across the vertical line \(x=4\) and then translated 2 units to the right.
Hint
Hint
Hint
Apply the reflection and then the translation. Remember that reflecting \(x\) across \(x=4\) changes \(x\) to \(8-x\).
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Answer
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- Formula Booklet
Complete Answer
\(g(x)= (6-x)^2 + 1\)
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Question
Determine the new coordinates of the vertex of \(f(x)= (x-4)^2 + 1\) after it has been transformed to \(g(x)\).
Hint
Hint
Hint
The vertex of \(f(x)\) is at (4, 1); apply the same transformations to this point.
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Answer
Answer
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- Formula Booklet
Complete Answer
The vertex of \(g(x)\) is at \((6, 1)\).
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Question
Discuss how the domain of \(f(x)= (x-4)^2 + 1\), originally defined for \(x\) in [0, 8], is affected by these transformations.
Hint
Hint
Hint
Consider the effect of the reflection on the endpoints and how the translation further adjusts them.
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- Formula Booklet
Complete Answer
After applying the transformations, the endpoints of the original domain [0, 8] are reflected and then translated, resulting in a new domain of [2, 10].
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